Folding the Biggest Box from a Sheet of Cardboard

Cut a square of side x from each corner of a sheet and fold up the sides. The box's volume is a cubic in factored form, x(L − 2x)(W − 2x), and the factors show at once where the volume is zero, which cuts make sense, and roughly where the best box hides.

How it works

Factored polynomials: zeros and graphs

In one sentence

A polynomial in factored form $a(x - x_1)(x - x_2)\cdots(x - x_n)$ has zeros exactly at $x_1, \dots, x_n$ . The graph crosses the x-axis at a zero of odd multiplicity and touches it at a zero of even multiplicity, and the leading term $ax^n$ sets the end behavior.

General formula

{\htmlData{sym=0}{V(x)}} = {\htmlData{sym=1}{x}}\,({\htmlData{sym=2}{L}} - 2{\htmlData{sym=1}{x}})\,({\htmlData{sym=3}{W}} - 2{\htmlData{sym=1}{x}})

A product is zero only when one of its factors is zero, so each factor marks a cut where the box collapses flat.

What the symbols mean

$V(x)$
Box volume
Cubic centimeters the folded box holds.
$x$
Corner cut
The side of the square cut from each corner, which becomes the box's height.
$L$
Sheet length
The long side of the cardboard.
$W$
Sheet width
The short side of the cardboard.

After cutting squares of side $x$ , the base is $(L - 2x)$ by $(W - 2x)$ and the height is $x$ , so $V(x) = x(L - 2x)(W - 2x)$ .
In factored form the zeros are visible: $x = 0$ , $x = W/2$ and $x = L/2$ . Each makes one dimension of the box zero.
Only $0 < x < W/2$ folds into a real box. Between the zeros the cubic's sign alternates, and its leading term $4x^3$ makes it rise on the far right.
The biggest box sits between the first two zeros. Calculus finds it exactly; the factored form already tells you where to look.

Learn the math

Factoring PolynomialsAlgebra · Lesson 5, Preliminaries Zeroes of PolynomialsAlgebra · Lesson 35, Polynomial Functions Graphing PolynomialsAlgebra · Lesson 36, Polynomial Functions